Expanding the Hyperbolic Kernels: A Curvature-aware Isometric Embedding View

Abstract

Modeling data relation as a hierarchical structure has proven beneficial for many learning scenarios, and the hyperbolic space, with negative curvature, can encode such data hierarchy without distortion. Several recent studies also show that the representation power of the hyperbolic space can be further improved by endowing the kernel methods. Unfortunately, the known kernel methods, developed in hyperbolic space, are limited by the adaptation capacity or distortion issues. This paper addresses the issues through a novel embedding function. To this end, we propose a curvature-aware isometric embedding, which establishes an isometry from the Poincar\'e model to a special reproducing kernel Hilbert space (RKHS). Then we can further define a series of kernels on this RKHS, including several positive definite kernels and an indefinite kernel. Thorough experiments are conducted to demonstrate the superiority of our proposals over existing-known hyperbolic and Euclidean kernels in various learning tasks, e.g., graph learning and zero-shot learning.